Cantor’s theorem
E78328
Cantor’s theorem is a fundamental result in set theory stating that the power set of any set has a strictly greater cardinality than the set itself, implying there is no largest infinity.
All labels observed (4)
| Label | Occurrences |
|---|---|
| Cantor’s theorem canonical | 3 |
| Cantor’s diagonal argument | 2 |
| Cantor theorem | 1 |
| Cantor's theorem | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T624849 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Cantor’s theorem Context triple: [Cantor’s paradox, usesConcept, Cantor’s theorem]
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A.
Cantor’s paradox
Cantor’s paradox is a foundational result in set theory showing that the “set of all sets” cannot exist because its power set would have a strictly larger cardinality, leading to a contradiction.
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B.
von Neumann paradox in set theory
The von Neumann paradox in set theory is a foundational result showing that, under certain group-theoretic conditions, a set can be decomposed and reassembled into paradoxical subsets of equal “size,” illustrating the counterintuitive consequences of the axiom of choice.
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C.
Burali-Forti paradox
The Burali-Forti paradox is a foundational logical contradiction in set theory that arises from considering the set of all ordinal numbers, showing that such a totality cannot consistently exist as a set.
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D.
Tarski's undefinability theorem
Tarski's undefinability theorem is a fundamental result in mathematical logic showing that, in sufficiently strong formal systems, the notion of truth for the language of the system cannot be defined within that same language.
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E.
Russell’s paradox
Russell’s paradox is a foundational logical contradiction in naive set theory that reveals problems with sets that contain themselves, leading to major developments in modern logic and the axiomatization of set theory.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Cantor’s theorem Target entity description: Cantor’s theorem is a fundamental result in set theory stating that the power set of any set has a strictly greater cardinality than the set itself, implying there is no largest infinity.
-
A.
Cantor’s paradox
Cantor’s paradox is a foundational result in set theory showing that the “set of all sets” cannot exist because its power set would have a strictly larger cardinality, leading to a contradiction.
-
B.
von Neumann paradox in set theory
The von Neumann paradox in set theory is a foundational result showing that, under certain group-theoretic conditions, a set can be decomposed and reassembled into paradoxical subsets of equal “size,” illustrating the counterintuitive consequences of the axiom of choice.
-
C.
Burali-Forti paradox
The Burali-Forti paradox is a foundational logical contradiction in set theory that arises from considering the set of all ordinal numbers, showing that such a totality cannot consistently exist as a set.
-
D.
Tarski's undefinability theorem
Tarski's undefinability theorem is a fundamental result in mathematical logic showing that, in sufficiently strong formal systems, the notion of truth for the language of the system cannot be defined within that same language.
-
E.
Russell’s paradox
Russell’s paradox is a foundational logical contradiction in naive set theory that reveals problems with sets that contain themselves, leading to major developments in modern logic and the axiomatization of set theory.
- F. None of above. chosen
Statements (43)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical theorem
ⓘ
result in set theory ⓘ |
| appliesTo |
every set
ⓘ
finite sets ⓘ infinite sets ⓘ |
| consequence |
for any infinite cardinal κ, 2^κ > κ
ⓘ
hierarchy of ever-larger infinities ⓘ no set is equinumerous with its power set ⓘ |
| coreConcept |
cardinality
ⓘ
infinite sets ⓘ power set ⓘ uncountability ⓘ |
| domain | foundations of mathematics ⓘ |
| field | set theory ⓘ |
| formalExpression |
∀S (|S| < |P(S)|)
ⓘ
∀S ¬∃f : S → P(S) such that f is surjective ⓘ |
| historicalPeriod | late 19th century ⓘ |
| holdsIn |
ZFC
ⓘ
linked to:
ZF
Zermelo–Fraenkel set theory ⓘ most standard axiomatic set theories ⓘ |
| implies |
the existence of infinitely many distinct infinite cardinalities
ⓘ
the real numbers are uncountable ⓘ the set of all subsets of the natural numbers is uncountable ⓘ there is no largest cardinal number ⓘ there is no largest infinity ⓘ |
| importance |
fundamental result in set theory
ⓘ
key to understanding different sizes of infinity ⓘ |
| influenced |
development of modern set theory
ⓘ
philosophy of mathematics ⓘ |
| isPartOf |
the study of cardinal numbers
ⓘ
the theory of infinite sets ⓘ |
| namedAfter | Georg Cantor ⓘ |
| proofIdea | assume a surjection from S to P(S) and derive a contradiction using a specially constructed subset ⓘ |
| relatedTo |
Cantor’s diagonal argument
ⓘ
linked to:
Cantor’s theorem
Cantor’s paradox ⓘ cardinal arithmetic ⓘ continuum hypothesis ⓘ |
| states |
for any set S, the power set P(S) has strictly greater cardinality than S
ⓘ
for every set S, |S| < |P(S)| ⓘ there is no surjection from a set S onto its power set P(S) ⓘ |
| status | proven ⓘ |
| usesMethod |
diagonal argument
ⓘ
proof by contradiction ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Cantor’s theorem Description of subject: Cantor’s theorem is a fundamental result in set theory stating that the power set of any set has a strictly greater cardinality than the set itself, implying there is no largest infinity.
Referenced by (7)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Cantor’s theorem
linked to: Cantor’s theorem
linked to: Cantor’s theorem
linked to: Cantor’s theorem
subject linked to:
Cantor